# What is a function that separates points of a manifold? [closed]

In the context of differential geometry, what is a function that separates points of a manifold?

## closed as off-topic by Saad, dantopa, Parcly Taxel, Lord Shark the Unknown, CesareoMar 16 at 9:03

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• Can you give an example of what you mean with two points on a plane in $\mathbb R^3 ?$ – Narasimham Mar 16 at 7:12

If you have a family $$\mathcal F$$ of functions from a set $$X$$ into some other set, we say that it separates the points of $$x$$ if, for each $$x_1,x_2\in X$$ with $$x_1\neq x_2$$, there is some $$f\in\mathcal F$$ such that $$f(x_1)\neq f(x_2)$$.
So, if $$\mathcal F$$ consists of a single function $$f$$, this is the same thing as asserting that $$f$$ is injective.
A function that separates points is a (smooth) real-valued function which has different value at the different points. If we only have two points, it is possible (depending on your lecturer and / or textbook author) that such a function is required to have the value $$0$$ at one point and $$1$$ at the other.
Let $$A$$ and $$B$$ be sets and let $$S$$ be a set of functions $$f:A \to B.$$ $$S$$ is said to separate the points of $$A$$, if for any $$x,y \in A$$ with $$x \ne y$$, there is $$f \in S$$ such that $$f(x) \ne f(y)$$.